We study the process of selective measurements of states of individual quantum systems - Josephson qubit – using nonlinear oscillator, working in the mesoscopic regime, when the number of quanta in the measuring process varies from a few dozen to a few hundred. Quantum Monte-Carlo method simulated dissipative dynamics of the system "qubit - oscillator" and the measurement process of a qubit state to modify the number of quanta of the oscillator. It is shown that for different Rabi-pulses of the recording state of a qubit the discrimination of states is possible, as well as the measurement of the effect of back-action of the measuring device, including separation of the prepared superposition state – carrying out statistical projective measurements.
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